Summary

DescriptionSpectral views of zero-fill and interpolation by lowpass filtering.svg	English: The first triangle of the first graph represents the Fourier transform X(f) of a continuous function x(t). The entirety of the first graph depicts the discrete-time Fourier transform of a sequence x[n] formed by sampling the continuous function x(t) at a low-rate of 1/T. The second graph depicts the application of a lowpass filter at a higher data-rate, implemented by inserting zero-valued samples between the original ones. And the third graph is the DTFT of the filter output. The bottom table expresses the maximum filter bandwidth in various frequency units used by filter design tools.
Date	3 January 2020
Source	Own work. The first and second graphs correspond to the first and third graphs of Harris[1], Figure 2.12, except the upsample rates are different (3 and 5)
Author	Bob K
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Other versions	This file was derived from: Spectral views of zero-fill and interpolation by lowpass filtering.pdf Category:Derivative versions
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References InfoField	↑ Harris, Frederic J. (24 May 2004) "2.2" in Multirate Signal Processing for Communication Systems, Upper Saddle River, NJ: Prentice Hall PTR, p. 23 ISBN: 0131465112.

Interpolation filter design

Let ${\displaystyle X(f)}$ be the Fourier transform of any function, ${\displaystyle x(t),}$ whose samples at some interval, ${\displaystyle T,}$ equal the ${\displaystyle x[n]}$ sequence. Then the discrete-time Fourier transform (DTFT) of the ${\displaystyle x[n]}$ sequence is the Fourier series representation of a periodic summation of ${\displaystyle X(f):}$

${\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ e^{-i2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}$

When ${\displaystyle T}$ has units of seconds, ${\displaystyle f}$ has units of hertz (Hz). Sampling ${\displaystyle L}$ times faster (at interval ${\displaystyle T/L}$) increases the periodicity by a factor of ${\displaystyle L:}$

${\displaystyle {\frac {L}{T}}\sum _{k=-\infty }^{\infty }X\left(f-k\cdot {\frac {L}{T}}\right),}$

which is also the desired result of interpolation. An example of both these distributions is depicted in the first and third graphs.

When the additional samples are inserted zeros, they decrease the sample-interval to ${\displaystyle T/L.}$ Omitting the zero-valued terms of the Fourier series, it can be written as:

${\displaystyle \sum _{n=0,\pm L,\pm 2L,...,\pm \infty }{}x(nT/L)\ e^{-i2\pi fnT/L},}$

which is equivalent to the first formula above, regardless of the value of ${\displaystyle L.}$ What ${\displaystyle L}$ determines is the DTFT periodicity of a digital filter implemented at the higher data-rate. The second graph depicts a lowpass filter and ${\displaystyle L=3,}$ resulting in the desired spectral distribution (third graph). The filter's bandwidth is the Nyquist frequency of the original ${\displaystyle x[n]}$ sequence. In units of Hz that value is ${\displaystyle {\tfrac {0.5}{T}},}$  but filter design applications usually require normalized units.